1. Introduction

2. Mathematical Model

2.1. SVPWM Technology

The core principle of Space Vector Pulse Width Modulation (SVPWM) involves managing the inverter’s output of three sinusoidal voltage signals by employing distinct combinations and sequences of activation and deactivation. These actions directly influence the stator windings of the PMSM. The resultant vector voltage, denoted as $U_{out}$, undergoes rotation along a predetermined path in accordance with the three-phase voltage signals. This rotational control effectively governs the PMSM in the locomotive, ensuring precise control over the motor to produce the desired torque output.

Figure 2. Linear combination of voltage space vector figure.

Figure 2. Linear combination of voltage space vector figure.

In Figure 1, the synthesis of voltage vectors is depicted. Focusing on sector 1, the resultant output voltage ($U_{\mathrm{out}}$) is composed by combining non-zero vector voltages $U_4$(1 1 0) and $U_6$(1 0 1) with the zero vector $U_0$(1 0 0) from two neighboring regions, accounting for their respective durations of influence.

In summary:

$$\left\{\begin{array}{l}{T}_4=\sqrt{3}\frac{U_{\mathrm{m}}}{U_{DC}}{T}_s\sin \left(\frac{\pi }{3}-{\theta }_i\right)\\ T_6=\sqrt{3}\frac{U_m}{U_{DC}}{T}_s\sin {\theta }_i\\ T_0=T_7=\frac{1}{2}\left(T_{\mathrm{s}}-T_4-T_6\right)\end{array}\right.$$

(9)

Where: $T_s$ - modulation time, $T_4$ - action time of $U_4$, $T_6$ - action time of $U_6$, $T_0$ - action time of $U_0$, $U_{out}$ - output voltage, $U_{DC}$ - DC bus side voltage, $U_m$ - RMS value of phase voltage, ${\theta }_i$ - the angle between the principal vector and the synthesized vector.

2.2. Vector Control of PMSM

The control strategy of PMSM is investigated as it is controlled to generate braking torque and control the locomotive parking by vector control method during locomotive parking.

$$T_e=T_{tq}=\frac{3}{2}{p}_n{i}_q{\psi }_f=\frac{\left(Gf\mathit{cos}\alpha +\frac{C_{D}A{v}_a^2}{21.15}+G\mathit{sin}\alpha -\frac{\delta m{v}^2}{2x}\right)r}{i_g{i}_0{\eta }_T}$$

(10)

Where: $T_{tq}$ - torque of motor, $i_g$ - ratio of locomotive transmission, $i_0$ - main reduction gear ratio of locomotive, ${\eta }_T$ - mechanical efficiency, $C_D$ - air resistance coefficient, A - front projection area, $v_a$ - traveling speed. G - gravity force on the locomotive (including self-weight and load), α - inclination angle, f - rolling resistance coefficient of the track, r - radius of the locomotive wheels, δ - rotating mass conversion factor of the locomotive, m - mass of the locomotive x - distance from target, v - different speeds.

q-axis current of the motor:

$$i_q=\frac{2\left(Gf\mathit{cos}\alpha +\frac{C_{D}A{v}_a^2}{21.15}+G\mathit{sin}\alpha -\frac{\delta m{v}^2}{2x}\right)r}{3i_g{i}_0{\eta }_T{p}_n{\psi }_f}$$

(11)

As can be seen from Eq. 10, the braking torque output from the PMSM can be controlled by controlling the q-axis current of the motor, which, combined with the parking control method, realizes the parking of the locomotive under different initial speeds, distances, slopes, and loads in the underground of the coal mine.

Figure 3. Schematic diagram of double closed-loop vector control of PMSM.

Figure 3. Schematic diagram of double closed-loop vector control of PMSM.

2.3. F-PID Vector Control of PMSM

The PID control methodology finds application in the locomotive parking process, showcasing its widespread utility in the realm of motor control. When overseeing the operation of the PMSM for locomotive propulsion, the initial step involves establishing the target torque value ($T_e^{*}$) for the PMSM within the PID control system. Subsequently, the actual torque ($T_e$) output of the motor is periodically measured and compared against the designated torque value ($T_e^{*}$). The resulting deviation between the set and actual torque values is then input into the PID controller, which has been fine-tuned with proportional, integral, and derivative coefficients. This input enables the PID controller to generate the necessary output, thereby completing the control of the entire system.

The transfer function of the PID is:

$$G\left(s\right)=K_p\left(1+\frac{1}{K_{i}s}+K_{d}s\right)$$

(12)

Where: $K_p$, $K_i$, $K_d$ are proportional, integral and differential time constants, respectively.

The differential regulator can be likened to incorporating a damper into the system, effectively mitigating overshooting and enhancing system stability. A substantial differential time constant contributes to slowing down the system’s response. Hence, it becomes imperative to judiciously determine the values of the three parameters $K_p$ , $K_i$, and $K_d$ in accordance with specific situations.

In the control system, fuzzy rules are primarily based on the deviation (e) between the given torque ($T_e^{*}$) and the actual output torque ($T_e$) of the permanent magnet synchronous motor. For significant errors (large e), increasing $K_p$ speeds up system convergence, while reducing Kd mitigates overshooting. Moderate errors (medium e) call for an intermediate value of $K_i$ to prevent substantial overshooting during system stops. Small errors prompt a focus on control system stability, suggesting a smaller $K_p$ value. Fuzzy control defines subsets for torque deviation (e) and deviation rate (ec) as {NB, NM, NS, ZO, PS, PM, PB} corresponding to {negatively large, negatively medium, negatively small, zero, positively small, positively medium, and positively large}. Post-processing maps these subsets to adjust the PID’s three-time constants within the range [-6, 6].

Table 1. $K_p$ Fuzzy Rules.

Table 1. $K_p$ Fuzzy Rules.

ec	e
	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	PS	ZO
NM	PB	PB	PM	PM	PS	ZO	ZO
NS	PM	PM	PM	PS	ZO	NS	NM
ZO	PM	PS	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	ZO	ZO	NS	NM	NM	NM	NB
PB	ZO	NS	NS	NM	NM	NB	NB

Table 2. ${\mathit{K}}_{\mathit{i}}$ Fuzzy rules.

Table 2. ${\mathit{K}}_{\mathit{i}}$ Fuzzy rules.

ec	e
	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NM	NM	NS	ZO	ZO	ZO
NM	NB	NM	NM	NS	ZO	ZO	ZO
NS	NM	NM	NS	ZO	ZO	PS	PS
ZO	NM	NS	NS	ZO	PS	PS	PS
PS	NM	NS	ZO	PS	PS	PM	PM
PM	NS	ZO	ZO	NS	PS	PM	PB
PB	ZO	ZO	PS	PS	PS	PB	PB

Table 3. ${\mathit{K}}_{\mathit{d}}$ Fuzzy rules.

Table 3. ${\mathit{K}}_{\mathit{d}}$ Fuzzy rules.

ec	e
	NB	NM	NS	ZO	PS	PM	PB
NB	PS	PS	ZO	ZO	ZO	PM	PB
NM	NS	NM	NM	NS	PM	PM	PM
NS	NB	NM	NM	NS	PM	PS	PM
ZO	NM	NM	NM	NS	PS	PS	PM
PS	NB	NB	NS	NS	PS	PS	PM
PM	NM	NB	NS	NS	ZO	PM	PB
PB	PS	ZO	ZO	ZO	ZO	PB	PB

Finally, three surface models with time constants varying with e and ec are obtained, as shown in Figure 4.

3. Simulation and Experiment

3.1. Simulation

This study utilizes the previous dynamic analysis and examination of the principles controlling fuzzy PID double-closed-loop vector control for PMSM as a foundation. It proceeds to develop a simulation model for locomotive parking using Simulink. The model presented in this study has three distinct components: the motor control simulation model, the locomotive dynamics simulation model, and the driver simulation model. The suggested parking control mechanism for locomotives is thoroughly examined through the utilization of this simulation model.

3.1.1. F-PID Control: Simulation for Electric Locomotives

Drawing upon the established principles of permanent magnet synchronous motor vector control theory and F-PID control theory, a meticulous simulation model of permanent magnet synchronous motor F-PID double closed-loop vector control was constructed. This comprehensive model meticulously captures the intricacies of the motor’s operation under the influence of F-PID double closed-loop vector control.

Figure 6. F-PID double closed-loop vector control system for PMSM.

Figure 6. F-PID double closed-loop vector control system for PMSM.

In this study, we present a comprehensive Simulink model of a mine electric locomotive that achieves precise control through a novel F-PID double closed-loop vector control and SVPWM strategy. The model delves into the locomotive’s dynamics and encompasses three crucial components: motor control, locomotive dynamics, and driver simulation. The F-PID controller, empowered by a fuzzy logic system, adaptively adjusts PID gains to guarantee optimal performance across diverse load and speed scenarios, ensuring precise motor speed and torque control even under challenging operating conditions.

3.1.2. Modeling Electric Locomotive Dynamics with Simulink

Simulink dynamic model of an electric locomotive is a comprehensive simulation tool that can be used to model the locomotive’s electrical, mechanical, and control systems as illustrated in Figure 7. The model is divided into three main subsystems: the motor control subsystem, the locomotive dynamics subsystem, and the driver subsystem. The motor control subsystem implements a F-PID double closed-loop vector control strategy. This control strategy is designed to provide precise control of the motor’s speed and torque, even under demanding operating conditions. F-PID controller uses a fuzzy logic system to adaptively tune the PID gains, ensuring optimal performance under varying load and speed conditions.

The locomotive dynamics subsystem captures the mechanical and electrical dynamics of the locomotive. The model includes components such as the motor, gearbox, wheels, and track. The model also takes into account factors such as aerodynamic drag and rolling resistance. The driver subsystem represents the human driver of the locomotive. The model takes into account the driver’s inputs, such as the throttle position and brake pedal position, and generates the corresponding control signals for the motor controller. The Simulink model can be used to simulate a wide range of operating conditions, including starting, stopping, running at constant speed, and climbing and descending gradients. The model can also be used to evaluate the performance of different control strategies and to optimize the locomotive’s design parameters.

3.1.3. Validation of Three-Phase Stator Current Control

The image shows the simulated results in Simulink for the three-term stator current of the PMSM. The Figure 8 represents the current flowing in each of the three stator phases (A, B, and C). The time axis is in seconds, and the current axis is in amps. The graphs show that the stator current is sinusoidal and well-balanced, indicating that the PMSM is operating properly. The current amplitude is also relatively low, which is desirable for efficiency reasons. The three-term stator current is a key variable in the control of a PMSM. It is used to control the speed and torque of the motor. The simulated results in the image show that the fuzzy PID double closed-loop vector control strategy is able to achieve precise control of the stator current, even under demanding operating conditions.

Specific Observations

• The three stator currents are sinusoidal and well-balanced, indicating that the PMSM is operating properly.
• The current amplitude is relatively low, which is desirable for efficiency reasons.
• The stator current is well-regulated, even under demanding operating conditions.

The simulated results in the image show that the F-PID double closed-loop vector control strategy can achieve precise control of the three-term stator current of a PMSM, even under demanding operating conditions. This is important for ensuring the efficient and reliable operation of the motor.

3.1.4. Electromagnetic Torque in Locomotive Stopping

This is shown in the image (Figure 9), where the electromagnetic torque curve (a) is positive during 0-5s, indicating that the motor is producing driving torque. After 5s, the electromagnetic torque becomes negative, indicating that the motor is producing brake torque. The acceleration curve (b) shows that the locomotive reaches its set speed of 0v after 5s. The speed curve (c) shows that the locomotive maintains its set speed of 0v after 5s.

3.1.5. Analyzing Stopping Control Error under Diverse Working Conditions

The graph illustrated in Figure 10 shows the simulation results of the error of the stopping control method under different working conditions. Each color represents a different working condition, with red representing the average error of the stopping control method, blue representing the average error of the stopping control method under different load conditions, green representing the average error of the stopping control method under different speed conditions, and yellow representing the average error of the stopping control method under different gradient conditions.

The results show that the error of the stopping control method is generally small under all working conditions. However, the error is slightly higher under high load conditions and high-speed conditions. The error is also slightly higher on gradients. Overall, the stopping control method is effective in controlling the stopping of the locomotive under a variety of working conditions. However, the error of the stopping control method should be considered when designing and operating locomotives. In the next section, the author will discuss about experimental setup.

3.2. Experiment

While building a locomotive parking test bench to simulate the forces on the locomotive under different speeds, distances, gradients and loads, it is necessary to choose suitable hardware to simulate the various forces on the locomotive during its traveling process, and to detect the driving state of the locomotive.

Introduction of the principle of the experimental bench, the selection of the hardware of the experimental bench, the design of the controller and the design of the software, etc. At last, the group carries out the parking experiments on the locomotives under different speeds, distances, slopes and loads, and then analyzes the results of the experiments and verifies the parking control method.

3.2.1. Principle and Selection of Test Bench

The experimental process is delineated in Figure 12, while the experimental setup is depicted in Figure 11. Before initiating the experiment, a thorough assessment of the site conditions is conducted, and power is supplied to the experimental bench. Subsequently, the host computer is activated, and the locomotive’s parameters are meticulously defined as per Table 4. Thereafter, the locomotive’s parking parameters, including parking speed, distance, gradient, and load, are carefully established.

Utilizing the controller, the host computer simulates the driving torque during locomotive acceleration and the braking torque during locomotive deceleration through the regulation of the permanent magnet synchronous motor. Concurrently, the magnetic powder brake generates the corresponding resistive torque to emulate the locomotive’s resistive forces. Moreover, the motor propels the flywheel, replicating the locomotive’s inertia torque. The controller simulates the locomotive’s resistance torque based on the rotational speed and torque measurements. Following computation, the host computer governs the PMSM to mimic the driving torque during acceleration and the braking torque during deceleration. Additionally, it controls the magnetic powder brake to produce the corresponding torque, replicating the locomotive’s resistive torque. Furthermore, the host computer drives the flywheel to simulate the locomotive’s inertial torque.

3.2.2. Comparative Result Analysis

The subsequent stage involves the selection of specific hardware for the experimental bench and the construction of the experimental setup for locomotive parking under varying speeds, distances, gradients, and loads. This is followed by the validation of the proposed parking control method through a series of rigorous experiments.

Figure 13. (a) Velocity graph at different initial speeds, (b) Velocity graph under different stopping distances, (c) Velocity graph under different track gradients, (d) Velocity graph under different loads.

Figure 13. (a) Velocity graph at different initial speeds, (b) Velocity graph under different stopping distances, (c) Velocity graph under different track gradients, (d) Velocity graph under different loads.

This academic article investigates the performance of a stopping control method under diverse conditions, specifically examining varying initial stopping speeds, track inclinations, and load weights. The experiments were conducted on an experimental bench, following parameters detailed in Table 5. The method’s effectiveness was evaluated by systematically analyzing experimental results. For initial stopping speed conditions, the test bench simulated normal locomotive traveling speed, accelerating to 1.7 m/s within 8.8 seconds. The stopping control method, executed at one-second intervals, gradually decreased the locomotive’s speed and the slope of the displacement curve during the parking process from 9.7 to 49.8 seconds.

In the assessment of track inclinations, an experimental platform underwent an acceleration phase, reaching 1.8 m/s at 6.2 seconds, replicating an electric locomotive’s initial speed. The subsequent stopping process, from 6.2 to 34.9 seconds, involved uniformly decelerated linear movements, with the method effectively decreasing speed and displacement curve slope. The stopping control approach was further evaluated under different load weights, with the test platform accelerating to 1.8 m/s at 8.6 seconds and subsequently experiencing a gradual decrease in speed from 8.6 to 36.4 seconds. This study demonstrates the effectiveness of the stopping control method for electric locomotives under diverse operating conditions. The method’s ability to adapt to different initial speeds, track inclinations, and load weights makes it a valuable tool for ensuring safe and efficient locomotive operation.

3.2.3. Error and Validation

Table 6 summarizes the stopping errors of an electric mine locomotive under different working conditions. The error is defined as the difference between the actual stopping distance and the target stopping distance. The results show that the stopping error is generally small, within ±0.3 m, under all working conditions. However, the error tends to increase with increasing initial stopping speed, distance, and track gradient. Inclusive, the results suggest that the electric mine locomotive is capable of accurate parking under a variety of working conditions.

Table 6 shows the error of an electric mine locomotive in different working conditions, including initial stopping speed, distance, load, and track gradient. The error is defined as the difference between the actual stopping distance and the target stopping distance.

The Figure 14 shows that the error of the electric mine locomotive is generally small, within ±0.3 m, under all working conditions. However, the error tends to increase with increasing initial stopping speed, distance, and track gradient. For example, at a distance of 20 m and a track gradient of 0 degrees, the error is around 0.2 m for an initial stopping speed of 1.8 m/s. However, the error increases to around 0.3 m for an initial stopping speed of 3 m/s. Similarly, at an initial stopping speed of 1.8 m/s and a track gradient of 0 degrees, the error is around 0.2 m for a distance of 20 m. However, the error increases to around 0.3 m for a distance of 30 m.

Figure 14 suggests that the electric mine locomotive is capable of accurate parking under a variety of working conditions. However, it is important to note that the error of the parking control system tends to increase with increasing initial stopping speed, distance, and track gradient.

4. Conclusion

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